Midpoint & Distance

Find the midpoint and the distance between two points.

The idea

The midpoint and distance formulas answer the two basic questions about the segment joining $(x_1, y_1)$ and $(x_2, y_2)$: where its middle lies, and how long it is. Both answers are arithmetic on the coordinates.

Theorem (Midpoint and distance formulas).

For points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint of the segment joining them is $\left(\frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}\right),$ and the distance between them is $d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}.$

The midpoint is the point halfway along the segment, and each of its coordinates is the average of the corresponding coordinates of the endpoints. Averaging places the point halfway across the horizontal gap and halfway up the vertical gap, and halfway in both directions is halfway along the segment.

For the distance, pass from the first point to the second by moving $\Delta x = x_2 - x_1$ horizontally and $\Delta y = y_2 - y_1$ vertically. The two moves are perpendicular, so they are the legs of a right triangle whose hypotenuse is the segment itself, and the Pythagorean theorem gives its length. The order of the two points does not matter: swapping them changes the sign of each gap, and squaring removes the sign.

Ways to work on it

• Walkthrough. The midpoint and distance formulas on clean points.
• Practice. Distance between points forming a Pythagorean triple.
• Hardest. Compute a distance and a midpoint with less convenient coordinates.

Not sure where to start? Take the ten-question placement test.